TY - JOUR

T1 - On Nonexistence of Baras-Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations

AU - Galaktionov, Victor A

PY - 2008

Y1 - 2008

N2 - The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B-1 subset of R-N, N >= 3, u(t) = Delta u + c/vertical bar x vertical bar(2)u in B-1 x (0, T), u vertical bar(partial derivative B1) = 0, in the supercritical range c > c(Hardy) = (N-2/2)(2) does not have a solution for any nontrivial L-1 initial data u(0)(x) >= 0 in B-1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u(n) (x, t)} of classical solutions corresponding to truncated bounded potentials given by V (x) = c/vertical bar x vertical bar(2) (bar right arrow) V-n(x) = min{c/vertical bar x vertical bar(2), N} (n >= 1) diverges; i.e., as n -> infinity, u(n)(x, t) -> +infinity in B-1 x (0, T). Similar features of "nonexistence via approximation" for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u(0)(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u(t) = Delta(vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(p-1) u/vertical bar x vertical bar(2), m >= 1, p > 1, and u(t) = -Delta(2)u + c/vertical bar x vertical bar(4)u , c > c(H) = [N(N-4)/4](2), N > 4.

AB - The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B-1 subset of R-N, N >= 3, u(t) = Delta u + c/vertical bar x vertical bar(2)u in B-1 x (0, T), u vertical bar(partial derivative B1) = 0, in the supercritical range c > c(Hardy) = (N-2/2)(2) does not have a solution for any nontrivial L-1 initial data u(0)(x) >= 0 in B-1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u(n) (x, t)} of classical solutions corresponding to truncated bounded potentials given by V (x) = c/vertical bar x vertical bar(2) (bar right arrow) V-n(x) = min{c/vertical bar x vertical bar(2), N} (n >= 1) diverges; i.e., as n -> infinity, u(n)(x, t) -> +infinity in B-1 x (0, T). Similar features of "nonexistence via approximation" for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u(0)(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u(t) = Delta(vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(p-1) u/vertical bar x vertical bar(2), m >= 1, p > 1, and u(t) = -Delta(2)u + c/vertical bar x vertical bar(4)u , c > c(H) = [N(N-4)/4](2), N > 4.

UR - http://www.scopus.com/inward/record.url?scp=43749085318&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1134/S0081543808010094

U2 - 10.1134/S0081543808010094

DO - 10.1134/S0081543808010094

M3 - Article

SN - 0081-5438

VL - 260

SP - 123

EP - 143

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

IS - 1

ER -